Pith. sign in

REVIEW 3 major objections 5 minor 2 cited by

Explicit symplectic integrators with adaptive time steps in curved spacetimes

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that a small symmetric modification of a time-transformed explicit symplectic integrator makes its physical step size adaptive while preserving the scheme's symplectic structure.

desk verdict The adaptive scheme is clever and the numerics look good, but the symplecticity argument fails: Eq. (16) treats an orbit-level total derivative as a Hamilton partial derivative, so the central claim is unproven. read the letter →

arxiv 2412.01045 v2 pith:BDWAEIHO submitted 2024-12-02 gr-qc astro-ph.IMmath-phmath.MP

Xin Wu , Ying Wang , Wei Sun , Fuyao Liu , Dazhu Ma This is my paper · ORCID
classification gr-qcastro-ph.IMmath-phmath.MP MSC 65P1037M1583C57
keywords explicitsymplecticintegratoradaptivetimestepstime-transformedHamiltoniancurvedspacetimesblackholegeodesicsSchwarzschild-Melvinsplittingbackwardraytracing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a small symmetric modification of an existing time-transformed explicit symplectic integrator makes its step size adapt to the local spacetime geometry while the symplectic character of the method is preserved. The modification introduces an auxiliary momentum $\Phi$ conjugate to the old time $\tau$, so the effective physical step becomes $d\tau = (r/j) g\, ds$, shrinking near a black-hole horizon and stretching at large radius. Compared with the nonadaptive scheme $S_2$, the new scheme $AS_2$ costs only two extra scalar updates per step and, in the reported tests, produces Hamiltonian errors typically two orders of magnitude smaller at the same step count. If the construction holds, adaptive stepping becomes available for every curved spacetime whose Hamiltonian, or time-transformed Hamiltonian, splits into explicitly integrable parts, with direct uses in long-term geodesic studies and black-hole ray tracing.

What carries the argument

The construction is carried by the split Hamiltonian $F = K/\Phi + g\ln(\Phi/\varphi)$ with $\varphi = j/r$, together with the composition $AS_2(h) = F_1(h/2)\, F_2(h/2)\, S_2(h/\Phi)\, F_2(h/2)\, F_1(h/2)$. Here $\Phi$ is the momentum conjugate to the old time $\tau$, treated as an additional coordinate; $F_1$ and $F_2$ are the flows of $-g\ln\varphi$ and $g\ln\Phi$, and $S_2$ is the existing second-order symmetric integrator built from explicitly integrable sub-Hamiltonians of $K$. The two logarithmic terms isolate the step-size control in two scalar flows, so the adaptive integrator needs only two extra updates per step, and the frozen-$\Phi$ solve of $S_2(h/\Phi)$ yields the time-step law $d\tau = (r/j) g\, ds$. This split is what makes adaptive stepping compatible with explicit symplectic integration instead of requiring an implicit solve.

What would settle it

Integrate a simple Schwarzschild geodesic with an independent high-accuracy solver and compare $d\Phi/ds$ from Eq. (24) with $-\partial F/\partial\tau = g\, d\ln(j/r)/d\tau$ evaluated from that independently computed $r(\tau)$; a systematic discrepancy at first order in the step would settle that the adaptive flow is not the Hamiltonian flow of $F$.

Watch

Extended reading notes

Core claim

The paper's central claim is that the extended-phase-space Hamiltonian $F = K/\Phi + g\ln(\Phi/\varphi)$, with $\varphi = j/r$, supports an explicit second-order symplectic integrator with adaptive steps in the original time. Writing $F_1 = -g\ln\varphi$ and $F_2 = g\ln\Phi$, the symmetric composition $AS_2(h) = F_1(h/2)\, F_2(h/2)\, S_2(h/\Phi)\, F_2(h/2)\, F_1(h/2)$ advances the spatial coordinates with the existing integrator $S_2$ while $\Phi$ and $\tau$ receive half-step scalar updates at the edges. $\Phi$ is frozen during the spatial solve and advanced by $\Phi(s) = \Phi_0 - s\, g\, g^{rr} p_r / r$, acting only as a rescaling of the time step, which keeps the implementation cheap. Because the step $h$ in the new time $s$ is fixed, the paper argues that the symplectic structure is preserved, while the old-time step $d\tau = (r/j) g\, ds$ varies with the orbit. Tests on Schwarzschild, Kerr, and Schwarzschild-Melvin spacetimes, for both particles and photons, show Hamiltonian errors about two orders smaller than $S_2$, and the two methods disagree on one Schwarzschild-Melvin orbit that $AS_2$ labels weakly chaotic.

Load-bearing premise

The load-bearing premise is that the orbit-level change of $\varphi = j/r$ can be treated as an explicit partial derivative of the Hamiltonian with respect to the time coordinate, as Eq. (16) does; if that step is not a legitimate Hamiltonian equation, the method's symplectic property is unproved.

Editorial extensions

If this is right

  • Long-term integrations of particle and photon geodesics in non-integrable spacetimes can use variable physical step sizes without giving up an explicit symplectic integrator.
  • For the Schwarzschild-Melvin parameters tested, $AS_2$ changes the inferred dynamics of one orbit from regular to weakly chaotic, so published portraits of chaos computed with nonadaptive $S_2$ may need re-examination near black-hole horizons.
  • The method inherits the applicability of $S_2$: any spacetime whose Hamiltonian, or time-transformed Hamiltonian, splits into explicitly integrable terms can use $AS_2$ with no structural changes.
  • Ray-tracing codes can choose $j$ near the observer distance, obtaining small steps near the photon sphere where shadow structure is decided and larger steps far away.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the derivation of Eq. (16) is legitimate, the same frozen-$\Phi$ trick should compose with higher-order symmetric integrators, giving adaptive versions of fourth- and sixth-order explicit symplectic schemes.
  • A practical extension would be an automatic or adaptive choice of $j$, since the paper's recommended range $(r_{\min}+r_{\max})/2 \le j \le r_{\max}$ is orbit-dependent and trading accuracy against the $\tau$-$s$ drift is currently a manual decision.
  • The $S_2$-versus-$AS_2$ disagreement on Orbit 1 is a testable warning: re-running earlier chaotic-transition scans near Schwarzschild-Melvin horizons with an adaptive scheme may shift apparent critical magnetic-field strengths.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes an explicit adaptive-time-step integrator AS2 for Hamiltonian geodesic motion in curved spacetimes. The construction combines the authors' earlier time-transformed explicit symplectic schemes with the Preto–Saha auxiliary momentum Φ: the Hamiltonian F = K/Φ + g ln(Φ/φ), with φ = j/r, is split as F = (1/Φ)Σ K_i + F1 + F2, and a symmetric composition AS2(h) = F1(h/2) F2(h/2) S2(h/Φ) F2(h/2) F1(h/2) is used with fixed step h in the new time s and adaptive steps in the old time τ. The paper claims that AS2 is symplectic in the extended phase space, costs only two extra scalar updates per step compared to S2, and typically gives two orders of magnitude smaller Hamiltonian errors. Tests are presented for Schwarzschild, Kerr, and Schwarzschild–Melvin spacetimes, including particle and photon orbits, and the method is applied to detect weak chaos in a Schwarzschild–Melvin orbit that appears regular under S2.

Significance. A simple, cheap, adaptive explicit symplectic integrator for curved spacetimes would be a valuable tool for long-term geodesic integrations, ray tracing, and chaos studies. The numerical demonstrations show consistent improvements in Hamiltonian-error size for the reported orbits, and the method is broad in principle because it inherits the splitting compatibility of the earlier S2 schemes. However, the central theoretical claim — that AS2 is symplectic and that the adaptive mechanism follows from Hamilton's equations — is not valid as derived, and the numerical evidence alone does not establish symplecticity. Since the title, abstract, and the main novelty of the paper rest on this claim, the contribution's central foundation needs to be reassessed.

major comments (3)
  1. [§2.2, Eq. (16)] The derivation of the adaptive mechanism is inconsistent with Hamiltonian mechanics. With φ = j/r, the Hamiltonian F in Eq. (15) has no explicit dependence on the coordinate τ because r and θ are independent phase-space coordinates; therefore ∂F/∂τ = 0 and Hamilton's equation gives dΦ/ds = 0, not the nonzero expression in Eq. (16). The manuscript obtains dΦ/ds = g d ln φ/dτ by differentiating φ along the orbit, which is a total derivative along the solution, not a partial derivative of F at fixed phase-space point. This invalidates Eqs. (19)–(21) and the crucial relation Φ/φ = 1, and hence the step-size adaptation mechanism itself is not a consequence of the Hamiltonian F.
  2. [§2.2, Steps 1–5 and Eq. (24)] The maps labeled F1(h/2) and F2(h/2) in Eq. (26) are not exact Hamiltonian flows of the sub-Hamiltonians defined by F1 = −g ln φ and F2 = g ln Φ. The implementation in Steps 1, 2, 4, and 5 freezes r, θ, and the momenta p_r, p_θ during the substeps governed by these terms, so that (for example) the update of Φ in Eq. (24) treats g, r, and p_r as constants. The exact flow of −g(r,θ) ln φ would also evolve r and θ through derivatives of g and φ. Consequently AS2(h) is a composition of non-Hamiltonian maps and is not a symplectic splitting of F; the assertion in Point 4 that 'the integrator AS2 remains symplectic' is therefore unsupported.
  3. [§2.2, Eq. (27)] The claimed old-time/new-time relation dτ = (r/j) g ds depends on the combination of Eq. (17) and Eq. (21). Since Eq. (21) relies on the invalid Eq. (16), the derivation of Eq. (27) does not follow from Hamilton's equations for F. Even if the adaptive update is retained as a time-step control heuristic, the paper does not provide a symplectic interpretation of the resulting map, and the method's good numerical energy behavior cannot be attributed to preservation of the canonical structure of F.
minor comments (5)
  1. [§1, paragraph after Eq. (14)] The phrase 'gives place to another form' should be 'gives way to another form,' and the sentence containing 'j ≥ rmax is possibly admitted' is unclear about whether j = rmax is allowed.
  2. [§2.2, footnote 5] Footnote 5 acknowledges the nontrivial relation between r, θ, and τ in φ, but the coordinate dependence is exactly the point that breaks Eq. (16); the footnote does not resolve the inconsistency and should be expanded or removed.
  3. [§3.3.1, Figure 5a] The orbits referred to as Orbit 1 through Orbit 7 are not explicitly defined in the text or figure; the description of which curves correspond to which initial radii would help reproducibility.
  4. [Tables 1 and 2] The tables list only orders of magnitude for the Hamiltonian error, without stating the norm or the exact time at which the error is measured (except for the final time); a precise definition would make the comparisons more reproducible.
  5. [References] The reference list contains formatting inconsistencies, such as 'Virbhadra1, K. S.' and 'Kop ´aˇcek, O.', and several entries lack complete page numbers or article numbers; these should be corrected before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the adaptive step-size law is a designed property of the constructed Hamiltonian F, not a fitted or self-cited prediction, and the numerical comparisons are externally anchored.

full rationale

The paper's derivation is a standard Hamiltonian-splitting construction: F is decomposed as K/Phi plus F1 and F2, and AS2 is the symmetric composition of explicitly stated sub-flow updates. No parameter is fitted to the target accuracy measurements; the free parameter j is varied in an honest parameter study (Tables 1-2), and the reported accuracy comparisons are numerical demonstrations rather than predictions derived from fitted inputs. The adaptive step-size relation dtau = (r/j)g ds (Eq. 27) is indeed an intended consequence of choosing phi = j/r (Eq. 22) and arranging Phi/phi = 1 (Eqs. 19-21); this is legitimate algorithm design, not a disguised fit or a renamed empirical result. Reliance on prior work by the same group (Wang et al. 2021; Wu et al. 2021, 2022) is for explicit time-transformation functions and integrable splits, which are external, parameter-free inputs with stated assumptions and are not used as a uniqueness argument to force the present method. The Schwarzschild-Melvin chaos result is also checked against an independent earlier study (Li & Wu 2019), providing external anchoring. The formal concern that Eq. (16) uses a total derivative along the orbit where a partial derivative at fixed phase-space point is required is a mathematical-rigor issue about symplecticity, not a circularity of the derivation chain.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on: (1) the standard geodesic Hamiltonian structure of the three spacetimes, stated in Section 2; (2) the splits and time-transformation functions inherited from the authors' prior work (Wu et al. 2021, 2022), which are parameter-free but referenced rather than re-derived; (3) the extended phase-space construction of Preto-Saha, with the paper's informal treatment of ∂F/∂τ in Eq. (16); and (4) the design choice φ = j/r, whose scale j is a free parameter that materially affects the accuracy gain (Tables 1 and 2). No new physical entities are introduced; Φ is borrowed from Preto & Saha (2009). The accuracy-improvement claim is contingent on choosing j in the recommended range, which is reported transparently.

free parameters (2)
  • j, scale parameter in φ = j/r = j = 100 (bound/falling), j = 1000 (escaping photon); recommended range (rmin+rmax)/2 ≤ j ≤ rmax
    Hand-tuned per orbit; controls the rescaled time factor 1/Φ = r/j in dτ = (r/j)g ds and the accuracy gain over S2 (Tables 1 and 2).
  • Step size h in the new time s = h = 1 (most tests), 0.1 (escaping photon), 0.001 (falling photon)
    Standard numerical step size, chosen by hand per test; listed for completeness since the accuracy and cost claims are made at specific h values.
assumptions (5)
  • domain assumption The geodesic motion in each spacetime is governed by the Hamiltonian H = (1/2) g^{αβ} p_α p_β of Eq. (2) with the given metric components.
    Standard Hamiltonian formulation of geodesic motion in general relativity; stated in Section 2, Eqs. (1)-(3).
  • domain assumption For each spacetime a time transformation function g exists such that K = g(H + p0) splits into l explicitly integrable sub-Hamiltonians K_i (Eqs. 6-7).
    Inherited from the authors' prior work (Wu et al. 2021, 2022; Wang et al. 2021a-c); the splits are parameter-free derivations but are referenced, not re-derived, and they are load-bearing for both S2 and AS2.
  • standard math The extended phase space with τ as a coordinate and Φ as its conjugate momentum, with F = K/Φ + g ln(Φ/φ), is a valid Hamiltonian whose dynamics implement the desired time rescaling.
    Standard construction from Preto & Saha (2009), but the paper's derivation of Eq. (16) treats ∂F/∂τ as an orbit-level total derivative, an informal step on which the adaptivity relies.
  • ad hoc to paper The constraint Φ/φ = 1 (Eq. 21) can be imposed initially and is preserved along the integration.
    Follows from Eqs. (19)-(20) within the paper's construction, but the choice φ = j/r and the constant C = 1 are design choices specific to this paper.
  • domain assumption The time transformation functions approach 1 (or a constant) for large r, so that g is approximately 1 far from the black hole and S2 uses nearly constant old-time steps.
    Stated as the design requirement in Section 2.1; Section 3.3.2 notes an exception for Schwarzschild-Melvin photons where 1.2 < g < 2.5.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Explicit symplectic integrators with adaptive time steps in curved spacetimes." pith.science (2026). https://pith.science/paper/BDWAEIHO

@misc{pith2026241201045,
  author       = {Pith},
  title        = {Pith review of: Explicit symplectic integrators with adaptive time steps in curved spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDWAEIHO}},
  note         = {Machine review of arXiv:2412.01045}
}
abstract

Recently, our group developed explicit symplectic methods for curved spacetimes that are not split into several explicitly integrable parts, but are via appropriate time transformations. Such time-transformed explicit symplectic integrators should have employed adaptive time steps in principle, but they are often difficult in practical implementations. In fact, they work well if time transformation functions cause the time-transformed Hamiltonians to have the desired splits and approach 1 or constants for sufficiently large distances. However, they do not satisfy the requirement of step-size selections in this case. Based on the step-size control technique proposed by Preto $\&$ Saha, the nonadaptive time step time-transformed explicit symplectic methods are slightly adjusted as adaptive ones. The adaptive methods have only two additional steps and a negligible increase in computational cost as compared with the nonadaptive ones. Their implementation is simple. Several dynamical simulations of particles and photons near black holes have demonstrated that the adaptive methods typically improve the efficiency of the nonadaptive methods. Because of the desirable property, the new adaptive methods are applied to investigate the chaotic dynamics of particles and photons outside the horizon in a Schwarzschild-Melvin spacetime. The new methods are widely applicable to all curved spacetimes corresponding to Hamiltonians or time-transformed Hamiltonians with the expected splits. Also application to the backwards ray-tracing method for studying the motion of photons and shadows of black holes is possible.

Figures

Figures reproduced from arXiv: 2412.01045 by the authors.

Figure 1
Figure 1. — Numerical comparisons in between two cases without magnetic field [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. — Numerical comparisons in between the two methods [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. — Numerical comparisons in between the two methods [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: — Same as Figure 3, but a photon orbit falling into the black hole with the parameters [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: — Numerical comparisons in simulations of particles around the Schwarzschild-Melvin black hole. The step [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: — Same as Figure 5a but only for the use of [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: — Poincare sections for the motions of photons near the Schwarzschild-Melvin black holes with several values ´ of the magnetic field. The method AS2 with the step size h = 1 is adopted. The other parameters are E = 0.995 and j = 100. The angular momentum is always give…
Figure 8
Figure 8. Figure 8: — Numerical comparisons between the methods [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Mutual Information for particle pair and its application to diagnose Chaos in Curved Spacetime

    gr-qc 2024-12 conditional novelty 5.0 of 10

    Mutual information between two nearby particle trajectories distinguishes regular from chaotic orbits in Schwarzschild and Kerr spacetimes, matching the fast Lyapunov indicator.

  2. Chaotic motion of the charged test particle in a Kerr-MOG black hole with explicit symplectic algorithms

    gr-qc 2024-12 conditional novelty 4.0 of 10

    Chaos grows with energy, magnetic field, and the MOG parameter, and shrinks with spin and angular momentum, for a charged particle in a magnetized Kerr-MOG black hole.

Reference graph

Works this paper leans on

81 extracted references · 78 canonical work pages · cited by 2 Pith papers

  1. [1]

    Y., & Sironi, L

    Bacchini, F., Ripperda, B., Chen, A. Y., & Sironi, L. 2018, ApJS, 237, 6

    2018

  2. [2]

    Y., & Sironi, L

    Bacchini, F., Ripperda, B., Chen, A. Y., & Sironi, L. 2019, ApJS, 240, 40

    work page 2019

  3. [3]

    T., & Cunningham, J

    Bardeen, C. T., & Cunningham, J. M. 1973, ApJ, 183, 237

    work page 1973

  4. [4]

    2024, Appl

    Blanes, S., Casas, F., & Escorihuela-Tom\` a s, A. 2024, Appl. Numer. Math. 204, 86

    work page 2024

  5. [5]

    2008, Bol

    Blanes, S., Casas, F., & Murua, A. 2008, Bol. Soc. Esp. Math. Apl., 45, 89

    work page 2008

  6. [6]

    2010, Bol

    Blanes, S., Casas, F., & Murua, A. 2010, Bol. Soc. Esp. Math. Apl., 50, 47

    work page 2010

  7. [7]

    Blanes, S., & Moan, P. C. 2002, JCoAM, 142, 313

    work page 2002

  8. [8]

    Brown, J. D. 2006, PhRvD, 73, 024001

    work page 2006

Show all 81 references
  1. [9]

    2024, Eur

    Cao, W., Wu, X., & Lyv, J. 2024, Eur. Phys. J. C, 84, 435

    2024

  2. [10]

    2014, PhRvD, 89, 084038

    Cisterna, A., & Erices, C. 2014, PhRvD, 89, 084038

    2014

  3. [11]

    2021, PhRvD, 103, 044020

    Contreras, E., Ovalle, J., & Casadio, R. 2021, PhRvD, 103, 044020

    2021

  4. [12]

    Cunha, P. V. P., Herdeiro, C. A. R., Radu, E., & R\' u narsson, H. F. 2015, PhRvL, 115, 211102

    2015

  5. [13]

    Cunha, P. V. P., Herdeiro, C. A. R., Radu, E., R\' u narsson, H. F., & Wittig, A. 2016, PhRvD, 94, 104023

    2016

  6. [14]

    R., & Shipley, J

    Dolan, S. R., & Shipley, J. O. 2016, PhRvD, 94, 044038

    2016

  7. [15]

    2019, ApJL, 875, L1

    EHT Collaboration, et al. 2019, ApJL, 875, L1

    2019

  8. [16]

    Emel'yanenko, V. V. 2007, Celest. Mech. Dyn. Astron., 98, 191

    2007

  9. [17]

    Ernst, F. J. 1976, J. Math. Phys., 17, 54

    1976

  10. [18]

    1986, JCM, 44, 279

    Feng, K. 1986, JCM, 44, 279

    1986

  11. [19]

    Gao, B., & Deng, X. M. 2021, Eur. Phys. J. C, 81, 983

    2021

  12. [20]

    1995, PhRvL, 74, 1276

    Garc\' i a, A., Galtsov, D., & Kechkin, O. 1995, PhRvL, 74, 1276

    1995

  13. [21]

    2023, Symmetry, 15, 1848

    He, G., Huang, G., & Hu, A. 2023, Symmetry, 15, 1848

    2023

  14. [22]

    Hou, Y., Zhang, Z., Yan, H., Guo, M., & Chen, B.2022, PhRvD, 106, 064058

    2022

  15. [23]

    2022,Universe, 8, 369

    Hu, A., & Huang, G.. 2022,Universe, 8, 369

    2022

  16. [24]

    2023,Symmetry, 15, 1094

    Hu, A., & Huang, G. 2023,Symmetry, 15, 1094

    2023

  17. [25]

    Huang, L., & Deng, X. M. 2024, PhRvD,109, 124005

    2024

  18. [26]

    Huang, Z., Huang, G., & Hu, A.2022, ApJ, 925, 158

    2022

  19. [27]

    U., Kumar, J., Walia, R

    Islam, S. U., Kumar, J., Walia, R. K., & Ghosh, S. G. 2023, ApJ, 943, 22

    2023

  20. [28]

    2023, Mathematics of Computation, 92, 251

    Jayawardana, B., & Ohsawa, T. 2023, Mathematics of Computation, 92, 251

    2023

  21. [29]

    2013, ApJ, 777, 170

    Johannsen, T. 2013, ApJ, 777, 170

    2013

  22. [30]

    Junior, H. C. D. L., Cunha, P. V. P., Herdeiro, C. A. R., & Crispino, L. C. B. 2021, PhRvD, 104, 044018

    2021

  23. [31]

    1992, General Relativity and Gravitation, 24, 729

    Karas, V., & Vokrouhlick\' y , D. 1992, General Relativity and Gravitation, 24, 729

    1992

  24. [32]

    Kawashima, T., Ohsuga, K., Takahashi, H. R. 2023, ApJ, 949, 101

    2023

  25. [33]

    Kerr, R. P. 1963, PhRvL, 11, 237

    1963

  26. [34]

    2018, ApJ, 853, 53

    Kop\' a c ek, O., & Karas, V. 2018, ApJ, 853, 53

    2018

  27. [35]

    2010, ApJ, 722, 1240

    Kop\' a c ek, O., Karas, V., Kov\' a r, J., & Stuchl\' i k, Z. 2010, ApJ, 722, 1240

    2010

  28. [36]

    2021, PhRvD, 103, 024021

    Kolo s , M., Tursunov, A., & Stuchl\' i k, Z. 2021, PhRvD, 103, 024021

    2021

  29. [37]

    U., & Ghosh, S

    Kumar, J., Islam, S. U., & Ghosh, S. 2022, JCAP, 2022, 032

    2022

  30. [38]

    2019, European Physical Journal Plus, 134, 96

    Li, D., & Wu, X. 2019, European Physical Journal Plus, 134, 96

    2019

  31. [39]

    2023, Universe, 9, 365

    Liu, C., & Wu, X. 2023, Universe, 9, 365

    2023

  32. [40]

    2024, Universe, 10, 277

    Lu, J., & Wu, X. 2024, Universe, 10, 277

    2024

  33. [41]

    Melvin, M. A. 1965, Phys. Rev. B, 139, 225

    1965

  34. [42]

    1997, Celest

    Mikkola, S. 1997, Celest. Mech. Dyn. Ast., 67, 145

    1997

  35. [43]

    1999, Celest

    Mikkola, S., & Tanikawa, K. 1999, Celest. Mech. Dyn. Ast., 74, 287

    1999

  36. [44]

    2013, New Astronomy, 20, 38

    Mikkola, S., & Tanikawa, K. 2013, New Astronomy, 20, 38

    2013

  37. [45]

    2023, SIAM Journal on Numerical Analysis, 61 (3), 1293

    Ohsawa, T. 2023, SIAM Journal on Numerical Analysis, 61 (3), 1293

    2023

  38. [46]

    2019, Eur

    P\' a nis, R., Kolo s , M., & Stuchl\' i k, Z. 2019, Eur. Phys. J. C, 79, 479

    2019

  39. [47]

    2022, MNRAS, 515, 1316

    Pelle, J., Reula, O., Carrasco, F., Bederian, C. 2022, MNRAS, 515, 1316

    2022

  40. [48]

    2015, Celestial Mech

    Pihajoki, P. 2015, Celestial Mech. Dynam. Astronom., 121, 211

    2015

  41. [49]

    2009, ApJ, 703, 1743

    Preto, M., & Saha, P. 2009, ApJ, 703, 1743

    2009

  42. [50]

    1999, AJ, 118, 2532

    Preto, M., & Tremaine, S. 1999, AJ, 118, 2532

    1999

  43. [51]

    2016, ApJ, 820, 105

    Pu, H., Yun, K., Younsi, Z., & Yoon, S. 2016, ApJ, 820, 105

    2016

  44. [52]

    Ruth, R. D. 1983, ITNS, 30, 2669

    1983

  45. [53]

    2012, PhRvD, 86, 124013

    Seyrich, J., & Lukes-Gerakopoulos, G. 2012, PhRvD, 86, 124013

    2012

  46. [54]

    Kolo s , M

    Stuchl\' i k, Z., M. Kolo s , M. 2016, Eur. Phys. J. C, 76, 32

    2016

  47. [55]

    2021a, Eur

    Sun, W., Wang, Y., Liu, F., & Wu, X. 2021a, Eur. Phys. J. C, 81, 785

  48. [56]

    2021b, Universe, 7, 410

    Sun, X., Wu, X., Wang, Y., Deng, C., Liu, B., & Liang, E. 2021b, Universe, 7, 410

  49. [57]

    2009, ApJ, 693, 472

    Takahashi, M., & Koyama, H. 2009, ApJ, 693, 472

    2009

  50. [58]

    R., Stein, L

    Tsang, D., Galley, C. R., Stein, L. C., & Turner, A. 2015, ApJL, 809, L9

    2015

  51. [59]

    2016, PhRvD, 93, 084012

    Tursunov, A., Stuchl\' i k, Z., & Kolo s , M. 2016, PhRvD, 93, 084012

    2016

  52. [60]

    2020, ApJ, 895, 14

    Tursunov, A., Stuchl\' i k, Z., Kolo s , M., Dadhich, N., & Ahmedov, B. 2020, ApJ, 895, 14

    2020

  53. [61]

    Virbhadra, K. S. 2024, PhRvD, 109, 124004

    2024

  54. [62]

    S., Narasimha, D., & Chitre, S

    Virbhadra1, K. S., Narasimha, D., & Chitre, S. M. 1998, Astron. Astrophys., 337, 1

    1998

  55. [63]

    Wald, R. M. 1974, PhRvD, 10, 1680

    1974

  56. [64]

    2021d, PhRvD, 104, 084021

    Wang, M., Chen, S., & Jing, J. 2021d, PhRvD, 104, 084021

  57. [65]

    2021a, ApJ, 907, 66

    Wang Y., Sun W., Liu F., Wu X. 2021a, ApJ, 907, 66

  58. [66]

    Wang Y., Sun W., Liu F., Wu X., 2021b, ApJ, 909, 22

  59. [67]

    2021c, ApJS, 254, 8

    Wang Y., Sun W., Liu F., Wu, X. 2021c, ApJS, 254, 8

  60. [68]

    White, C. J. 2022, ApJS, 262, 28

    2022

  61. [69]

    1982, AJ, 87, 577

    Wisdom, J. 1982, AJ, 87, 577

    1982

  62. [70]

    1991, AJ, 102, 1528

    Wisdom, J., & Holman, M. 1991, AJ, 102, 1528

    1991

  63. [71]

    Wu, X., Wang, Y., Sun, W., & Liu, F. Y. 2021, ApJ, 914, 63

    2021

  64. [72]

    Y, & Han, W

    Wu, X., Wang, Y., Sun, W., Liu, F. Y, & Han, W. B. 2022, ApJ, 940, 166

    2022

  65. [73]

    2022, Universe, 8, 320

    Yang, D., Cao, W., Zhou, N., Zhang, H., Liu, W., & Wu, X. 2022, Universe, 8, 320

    2022

  66. [74]

    2023, Eur

    Yang, D., Liu, W., & Wu, X. 2023, Eur. Phys. J. C, 83, 357

    2023

  67. [75]

    2023, Eur

    Yang, D., & Wu, X. 2023, Eur. Phys. J. C, 83, 789

    2023

  68. [76]

    1990, Phys

    Yoshida, H. 1990, Phys. Lett. A, 150, 262

    1990

  69. [77]

    2023, ApJ, 942, 47

    Younsi, Z., Psaltis, D., \" O zel, F. 2023, ApJ, 942, 47

    2023

  70. [78]

    2022, Eur

    Zhang, J., & Xie, Y. 2022, Eur. Phys. J. C, 82, 854

    2022

  71. [79]

    2021, Universe, 7, 488

    Zhang, H., Zhou, N., Liu, W., & Wu, X. 2021, Universe, 7, 488

    2021

  72. [80]

    2022, General Relativity and Gravitation, 54, 110

    Zhang, H., Zhou, N., Liu, W., & Wu, X. 2022, General Relativity and Gravitation, 54, 110

    2022

  73. [81]

    2022, ApJ, 927, 160; 2023, ApJ, 947, 94

    Zhou, N., Zhang, H., Liu, W., Wu, X. 2022, ApJ, 927, 160; 2023, ApJ, 947, 94

    2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.